Question
If a line (parallel, perpendicular) to one side of a triangle intersects the other two sides, then it divides the two sides (equally, proportionally).
Answers
GPT-4o mini
If a line is drawn parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally. This is a result of the Basic Proportionality Theorem (also known as Thales' theorem).
Specifically, if we have a triangle \(ABC\) and a line parallel to side \(BC\) that intersects sides \(AB\) and \(AC\) at points \(D\) and \(E\) respectively, then the following proportionality holds:
\[
\frac{AD}{DB} = \frac{AE}{EC}
\]
This means that the segments created on the two intersected sides are proportional to each other, not necessarily equal. Thus, the segments are divided in the same ratio.
Specifically, if we have a triangle \(ABC\) and a line parallel to side \(BC\) that intersects sides \(AB\) and \(AC\) at points \(D\) and \(E\) respectively, then the following proportionality holds:
\[
\frac{AD}{DB} = \frac{AE}{EC}
\]
This means that the segments created on the two intersected sides are proportional to each other, not necessarily equal. Thus, the segments are divided in the same ratio.